Use the divergence theorem to calculate the flux of the vector field F(x,y,z)=(x^3)i+(y^3)j+(z^3)k out of the closed, outward-oriented surface S bounding the solid x^2 + y^2

Question

Use the divergence theorem to calculate the flux of the vector field F(x,y,z)=(x^3)i+(y^3)j+(z^3)k out of the closed, outward-oriented surface S bounding the solid x^2 + y^2 <= 16 and 0<= z <= 7.

tiffany_rhodes · Answer

I calculated the divergence of F to be: 3x^2 + 3y^2 + 3z^2 = 3(x^2+y^2+z^2). I really just need help understanding whether I should spherical coordinates, etc. Once I get the problem set up, I know how to integrate.

tiffany_rhodes · Answer

$$3*\int\limits_{?}^{?} \int\limits_{?}^{?}\int\limits_{?}^{?} x^2 + y^2 + z^2 dx dy dz$$

inkyvoyd · Answer

Well, what's the shape of solid?

tiffany_rhodes · Answer

A solid cylinder?

inkyvoyd · Answer

ya, what kind of coordinate system do you think would make that easiest? ;)

tiffany_rhodes · Answer

ooh, cylindrical coordinates? :P

inkyvoyd · Answer

yeah... as far as I can tell... if you look at the integrand, you also get a nice expression... that is usually how I get an idea of what coordinate system to use. Don't forget to add the jacobian term r :)

tiffany_rhodes · Answer

Okay, I'll try that and see what I get. Thanks :)

inkyvoyd · Answer

good luck!